Automatic Threshold Detection for Tachometer Signals

ABSTRACT

A method is described for automatically determining a proper threshold for a tachometer signal in order to produce desired tachometer pulses necessary for analysis of machine vibration data. A tachometer signal is low-pass filtered to exclude high frequency noise and a running derivative of the filtered tachometer waveform is taken to create a derivative waveform. Another waveform is created that includes only positive values from the derivative waveform that correspond to positive values in the low-pass filtered tachometer waveform. In general, a tachometer signal has the greatest derivative value (slope) when a tachometer pulse is present. Based on this observation, a threshold value is determined using both the low-pass filtered tachometer waveform and the positive-value derivative waveform along with statistics from both waveforms.

RELATED APPLICATIONS

This nonprovisional patent application claims priority to U.S.provisional patent application Ser. No. 61/835,684, filed Jun. 17, 2013,titled “Automatic Threshold Detection for Tachometer Signals,” theentire contents of which are incorporated herein by reference.

FIELD

This invention relates to analysis of machine vibration data. Moreparticularly, this invention relates to a method for automaticallydetermining a proper threshold for a tachometer signal in order toproduce the desired tachometer pulses necessary for analysis of machinevibration data.

BACKGROUND

One of the most important characteristics of machine operation used inthe analysis of vibration data is the rotational speed of the machine.Machine speed is often acquired from tachometer measurements. Accuratetachometer pulses are essential for applications such as order tracking,synchronous time averaging, single channel phase, Bode plots, andgeneral turning speed calculations. Ideally, tachometer pulses aregenerated when illumination from a tachometer sensor passes over asection of reflective tape placed on a rotating portion of the machine,such as a rotating shaft. However, the shafts of most machines inindustry do not have reflective tape. If they have reflective tape,often times the tape is covered with grime or has rubbed off or isotherwise damaged to some extent. Because a machine typically cannot bestopped in order to place reflective tape on the shaft, an analyst willcollect tachometer data in hopes of getting a trigger from a keyway, ascratch on the shaft, or remnants of old reflective tape. The resultingtachometer waveform is typically noisy and small in amplitude and thetachometer pulses are not very distinctive.

An ideal tachometer waveform consists of a series of distinct pulses,wherein each pulse indicates a single revolution of the shaft. Togenerate a tachometer pulse sufficient for the applications mentionedabove, a threshold is set such that a rising (or falling) edge of thetachometer signal triggers a pulse when it passes the threshold. Thisthreshold is manually entered by an analyst. In an ideal situation,where reflective tape is present resulting in a strong tachometersignal, the chosen threshold value will generally be acceptable for allmeasurements. However, when reflective tape is not present, the level ofthe tachometer signal varies from one machine to the next and thenecessary threshold will be different for each measurement.

What is needed, therefore, is an automatic method for calculating aproper threshold for a given tachometer signal in order to produce thedesired tachometer pulses needed for machine vibration analysis.

SUMMARY

The various embodiments of the invention described herein extract anevent from data. The data of interest is generally characterized by arapid change in a signal amplitude. By applying a derivative to thesignal, these rapid changes in amplitude become more evident and aremuch easier to see and quantify. When evaluating tachometer signalsassociated with a machine being monitored, the rapid change in signalamplitude occurs at a rate substantially equivalent to the running speedof the machine. Such tachometer signals may be generated by varioussources, including but not limited to laser, optical, eddy current,magnetic, and LED sources.

Embodiments of the invention can also be applied to vibration signalshaving repetitive and impulsive features. One example of an impulsivesignal is a signal associated with a bearing defect. By taking thederivative of an acceleration waveform (sometimes referred to as a“jerk”), a threshold value can be calculated using an embodiment of theinvention that extracts a repetitive signal associated with theimpulsive characteristic generated from a defective bearing. Embodimentsof the invention may be applied to any vibration waveform to extractinformation associated with impulse-driven information.

A preferred method embodiment begins by passing a tachometer signalthrough a low-pass filter to exclude high-frequency noise. Preferably,the low-pass corner frequency of the filter is 250 Hz, but depending onthe signal the desired corner frequency may vary between 100 Hz and 1000Hz. Next, a running derivative of the filtered tachometer waveform istaken. The resulting signal is referred to herein as the derivativewaveform (“Deriv_WF”). A new waveform is then created that includes onlypositive values from the derivative waveform that correspond to positivevalues in the associated low-pass filtered tachometer waveform. This newwaveform is referred to herein as Deriv>0_WF. In general, a tachometersignal has the greatest derivative value (slope) when a tachometer pulseis present. Based on this observation, preferred embodiments of themethod derive a threshold value using both the low-pass filteredtachometer waveform and Deriv>0_WF along with statistics from bothwaveforms. In alternative embodiments, a Deriv<0_WF waveform is derivedusing only negative values in the associated low-pass filteredtachometer waveform. A negative threshold value may be calculated fornegative going pulses such as generated by proximity probe tachometers.

In preferred embodiments, determination of the tachometer thresholdvalue is based on finding a specified number of peaks in the Deriv>0_WFwaveform and locating their associated peaks in the low-pass filteredtachometer waveform. Using a combination of statistics derived from bothwaveforms, one of three options will be selected:

-   -   (1) a threshold is determined for the low-pass filtered        waveform, which in turn is used to produce tachometer pulses;    -   (2) although the low-pass filtered waveform does not provide a        sufficient signal to produce a tachometer pulse, the associated        Deriv>0_WF waveform does provide the information needed and        therefore a threshold is calculated and applied to the        Deriv>0_WF waveform that is used to generate tachometer pulses;

(3) neither the low-pass filtered waveform nor Deriv>0_WF contains thenecessary characteristics to produce a reliable tachometer pulse and theanalyst is warned that the acquired signal is insufficient to generatetachometer pulses.

Preferred embodiments of the method include:

-   -   (1) three different methods to determine peaks in the        Deriv>0_WF;    -   (2) the ability to sort out peak values that are not consistent        with peak values associated with the running speed;    -   (3) determination of the threshold value based on various        statistical criteria.

Peaks representative of tachometer pulses in the raw tachometer waveformare determined from the peaks found in the Deriv>0_WF waveform. TheDeriv>0_WF waveform consists of subsets or groups of positive peakvalues. (For example, in FIG. 4, such a group (or subset) contains allthe peaks found between “A” and “E.” Another group contains the singlepeak between “F” and “G.”) In some embodiments of the method, the peaksin Deriv>0_WF are found as the first peak of a group of positive peakvalues. In alternative embodiments, the selected peaks in Deriv>0_WFcorrespond to either the largest derivative value or the last peak fromeach group of peak values. To ensure that the selected set of Deriv>0_WFpeaks are associated only with the speed of the machine, outlying peaksare culled out using a sorting routine described hereinafter. Thisroutine preferably includes four methods to determine the selected peaksbased on calculations of the mean and standard deviation of the set ofpeaks.

Alternative embodiments of the method may employ a different statisticalprocess, such as a mode, to find a frequently recurring family of peakswithin the derivative waveform. For example, a statistical histogram, acumulative distribution, or another probability density technique may beimplemented to detect and identify a range of interest based on a modeor a most frequently recurring subset of data values within a populationof data values from the derivative waveform.

BRIEF DESCRIPTION OF THE DRAWINGS

Further advantages of the invention are apparent by reference to thedetailed description in conjunction with the figures, wherein elementsare not to scale so as to more clearly show the details, wherein likereference numbers indicate like elements throughout the several views,and wherein:

FIG. 1 depicts a functional block diagram of a system for derivinginformation from a tachometer signal according to an embodiment of theinvention;

FIGS. 2 and 3 depict flowcharts of steps of a method for determining athreshold for a tachometer signal according to a preferred embodiment ofthe invention; and

FIG. 4 depicts a portion of an example filtered tachometer waveform anda corresponding running derivative waveform.

DETAILED DESCRIPTION

FIG. 1 depicts a system 100 for deriving information from a tachometersignal. In the embodiment of FIG. 1, a tachometer 104 is attached to amachine 102 to monitor the rotational speed of a component of themachine 102, such as a rotating shaft. The tachometer 104 generates atachometer signal that contains information about the rotational speedof the machine 102. The tachometer signal is provided to a datacollector 106 comprising an analog-to-digital converter (ADC) 108 forsampling the tachometer signal, a low-pass filter 110, and buffer memory112. The data collector 106 may be a digital data recorder manufacturedby TEAC or a vibration data collector. In a preferred embodiment, theADC 108 samples the tachometer signal at 48,000 samples/second. Thelow-pass filter 110 is preferably an FIR filter with 49 taps, preferablywith a corner frequency set between 250 and 1000 Hz. In a preferredembodiment, the low-pass corner frequency is set at 250 Hz forrotational speeds of 900 RPM and higher.

In the embodiment of FIG. 1, the tachometer signal data is transferredfrom the data collector 106 to a threshold processor 114 that performsthe calculations and other information processing tasks describedherein. In an alternative embodiment, the calculations and processingare performed by a processor in the data collector 106.

As depicted in FIG. 2, a preferred method 10 for determining a thresholdfor a tachometer signal begins with the collection of the raw tachometerdata (step 12). The tachometer signal data is passed through thelow-pass filter 110 (step 14), which generates a filtered signal at itsoutput referred to herein as “Filtered_WF.”

Once Filtered_WF is stable (after about the 50^(th) sample), theroot-mean-square (RMS) value of Filtered_WF is calculated (step 16), andthe waveform is centered about the RMS value and then centered aboutzero with the RMS offset (DC offset) (step 18). The Crest Factor (CF)for Filtered_WF is then calculated according to:

$\begin{matrix}{{C\; F} = {\frac{{Largest}\mspace{14mu} {peak}\mspace{14mu} {of}\mspace{14mu} {Filtered\_ WF}}{{RMS}\mspace{14mu} {of}\mspace{14mu} {Filtered\_ WF}}.}} & \left( {{step}\mspace{14mu} 20} \right)\end{matrix}$

This value is referred to herein as Filtered_WF_CF_RMS.

A running derivative of Filtered_WF is taken which is referred to as theDeriv_WF signal (step 22), wherein the zero value derivative point atthe beginning is dropped. The Crest Factor (CF) for the Deriv_WF signalis calculated according to:

$\begin{matrix}{{C\; F} = {\frac{{Largest}\mspace{14mu} {peak}\mspace{14mu} {of}\mspace{14mu} {Deriv\_ WF}}{{RMS}\mspace{14mu} {of}\mspace{14mu} {Deriv\_ WF}}.}} & \left( {{step}\mspace{14mu} 24} \right)\end{matrix}$

This value is referred to as Deriv_CF_RMS.

Using the Deriv_WF signal, a new waveform is created from the positivevalues in Filtered_WF that have only the positive derivative values(slopes). The new waveform retains all values that are greater than zerowhile setting all negative values (i.e. slope≦0) to zero (step 26). Theresulting signal, referred to herein as Deriv>0_WF, looks like a“rectified” version of Deriv_WF, although this waveform is not actuallyrectified. Specifically, values greater than zero in Filtered_WF aredetermined, the slope of the resulting signal is calculated, and anyslopes that are less than or equal to zero are set to zero. The meanvalue (μ) of the resulting signal is found and referred to asDeriv>0_Mean, and the standard deviation (σ) is found and referred to asDeriv>0_SD (step 28). In the preferred embodiment, values of zero arenot used to calculate Deriv>0_Mean and Deriv>0_SD. In some embodiments,Deriv>0_WF is created in an opposite manner using only the negativeslope of Deriv_WF, and the absolute value is taken of the resultingsignal for further analysis.

The upper portion of FIG. 4 depicts an example of Deriv>0_WF showing aset of peaks starting after 0.0075 seconds (point “A”) and ending justafter 0.01 seconds (point “E”). The lower portion of FIG. 4 is thecorresponding Filtered_WF that was used to produce Deriv>0_WF in theupper portion. The vertical lines show the derivative (slope) valuesassociated with points in Filtered_WF.

In a preferred embodiment, all peaks in Deriv>0_WF are found by one ofthree methods (step 30). Each method evaluates every group of pointsdefined as a set of points (positive derivatives) bounded by zero:

-   -   (1) The first peak in each set is evaluated. In this method, the        peak is the first peak of the set where the peak is the most        positive value before becoming less positive. For example, see        point “B” in FIG. 4.    -   (2) Choose the “steepest” slope or largest derivative. This will        be the tallest peak in each set. The tallest peak is associated        with the steepest slope in the set. For example, see point “C”        in FIG. 4.    -   (3) Choose the last peak in the set. This point is associated        with the steepest slope of the line just before the peak occurs        in Filtered_WF. For example, see point “D” in FIG. 4.

The N number of largest peaks are then found in Deriv>0_WF (step 32).The value of N can be user-selected or calculated as described in aprocess performed in a preferred embodiment to calculate the number ofpeaks for analysis. Typically, N is 20 for one second of data at speedsgreater than 1800 RPM. The largest of these N peaks is found and theCrest Factor (CF) for the Deriv>0_WF data is determined according to:

$\begin{matrix}{{C\; F} = {\frac{{{Largest}\mspace{14mu} {of}\mspace{14mu} N\mspace{14mu} {peaks}\mspace{14mu} {of}\mspace{14mu} {Deriv}} > {0{\_ WF}}}{{{RMS}\mspace{14mu} {of}\mspace{14mu} {Deriv}} > {0{\_ WF}}}.}} & \left( {{step}\mspace{14mu} 34} \right)\end{matrix}$

This value is referred to as Deriv>0_CF_RMS. When calculating the RMSvalue of the Deriv>0_WF, values of zero are included.

The N peaks found in step 32 are sorted by amplitude from largest tosmallest (step 36), and any of the N peaks greater than the boundary ofμ+σ are discarded as outliers (step 38). A statistical method used in apreferred embodiment to discard the outlier data is described inhereinafter (method 2). A Crest Factor for the remaining values iscalculated according to:

$\begin{matrix}{{C\; F} = {\frac{\begin{matrix}{{{Largest}\mspace{14mu} {of}\mspace{14mu} N\mspace{14mu} {peaks}\mspace{14mu} {of}\mspace{14mu} {Deriv}} >} \\{0{\_ WF}\mspace{14mu} {excluding\_ outliers}}\end{matrix}}{{{RMS}\mspace{14mu} {of}\mspace{14mu} {Deriv}} > {0{\_ WF}{\_ excluding}{\_ outliers}}}.}} & \left( {{step}\mspace{14mu} 40} \right)\end{matrix}$

This value is referred to as Adj_Deriv>0_CF_RMS. Preferably, the RMScalculation used for the Adj_Deriv>0_CF_RMS value does not incorporatethe discarded peak values.

Next locations in Deriv>0_WF are determined where the signal crosseszero to the right of the peaks (FIG. 4, points “E” and “G”) (step 42).These locations coincide with a peak in the Filtered_WF signal. The“base” value of Deriv>0_WF is then determined for each of the N peaksfound in step 32 (step 44). This base value is preferably associatedwith the zero crossing of the Filtered_WF signal as indicated by point“A” in FIG. 4 or at a value where the derivative (slope) changes fromnegative to positive, such as point “F” in FIG. 4 which corresponds to avalley in the Filtered_WF data where the derivative changes fromnegative to positive.

In preferred embodiments, the analysis used for threshold calculationsis based on Filtered_WF data using either of two methods:

-   -   (1) Using the difference between the Filtered_WF peak values        determined in step 42 and the Filtered_WF values associated with        the “base” value of Deriv>0_WF found in step 44. This difference        is referred to as “Max_Diff” (step 46).    -   (2) Using the Filtered_WF peak values associated with the        Deriv>0_WF values found in step 42. This value is referred to as        “Max Value” (step 48).

The ratio of the CF values found in steps 40 and 20 is expressed as:

$\begin{matrix}{{{{Adj}/{Filter\_ CF}}{\_ RMS}} = {\frac{\left( {{Adj\_ Deriv} > {0{\_ CF}{\_ RMS}}} \right)}{\left( {{Filtered\_ WF}{\_ CF}{\_ RMS}} \right)}.}} & \left( {{step}\mspace{14mu} 50} \right)\end{matrix}$

If Adj/Filter_CF_RMS is greater than or equal to three (step 52) and nomaximum peaks were discarded in Deriv>0_WF (step 53), then theDeriv>0_WF waveform is used as the signal from which tachometer pulsesare derived. The tachometer signal threshold (step 54) is calculatedfrom the Deriv>0_WF waveform. If maximum peaks were discarded inDeriv>0_WF (step 53), then a “bad data” indication is generated (step55). If Adj/Filter_CF_RMS is less than three (step 52), then theFiltered_WF waveform is used to create tachometer pulses and as a basisto set the tachometer signal threshold (step 56). Whichever waveform isused to set the threshold is referred to herein as the “decision WF.”

Calculation of Threshold Value

The set of amplitudes (values) of the decision WF used to calculate thethreshold limit are preferably within μ±2σ, where the mean and standarddeviation are calculated from the set of values used in calculatingMax_Diff and Max_Value (steps 46 and 48). Details of a statisticalmethod for discarding data outside the limits is described hereinafter(Method 3 with n=2).

For calculations related to Max_Diff:

Max_Threshold=MaxΔ+Base Value (step 58); and

Min_Threshold=MinΔ+Base Value (step 60)

where

-   -   Δ=change (difference value) from the peak in the Filtered_WF        waveform and the closest left-most base value; and    -   Base value=amplitude value at a position in the Filtered_WF        waveform where the slope changes from≦zero to positive.        Min_Threshold can be greater than Max_Threshold because the MaxΔ        value could have a small base value compared to MinΔ which could        have an associated large base value.

For calculations related to Max_Value:

Max_Threshold=Amplitude of the largest peak taken from the sorted “N”peaks extracted from Filtered_WF (step 36) with Base Value=0 (step 62);and

Min_Threshold=Amplitude of the smallest peak taken from the sorted “N”peaks extracted from the Filtered_WF (step 36) with Base Value=0 (step64).

Calculation of the Percent Difference Threshold (referred to herein as%_Diff_Threshold) indicates how much the difference between two valueschange from the average:

$\begin{matrix}{{{Percent}\mspace{14mu} {Difference}\mspace{14mu} {Threshold}} = {\frac{({Max\_ Threshold}) - ({Min\_ Threshold})}{\left( \frac{({Max\_ Threshold}) + ({Min\_ Threshold})}{2} \right)}.}} & \left( {{step}\mspace{14mu} 66} \right)\end{matrix}$

The Percent Mean Filtered Max Peak (referred to herein as%_Mean_Fltd_Max_PK) is the percent change in the amplitude values of themaximum peaks in the Filtered_WF data:

$\begin{matrix}{{\% {\_ Mean}{\_ Fltd}{\_ Max}{\_ PK}} = {\left\lbrack {1 - \left( \frac{\mu - \sigma}{\mu} \right)} \right\rbrack*100{\%.}}} & \left( {{step}\mspace{14mu} 68} \right)\end{matrix}$

This parameter is the percent mean taken from the “N” maximum peaks ofthe Filtered_WF data, where N is either a user-selected number of peaksor is calculated as described hereinafter in a process that calculatesthe number of peaks for analysis. The mean and standard deviation arecalculated for the set of maximum peak values taken from the Filtered_WFdata.

If %_Mean_Fltd_Max_PK from step 68 is larger than ten, then the data isconsidered “questionable.” This means the data “jumps” around too muchand it is difficult, if not impossible, to set a realistic threshold. Athreshold can still be calculated but will probably not be useful.

As shown in FIG. 3, if Adj/Filter_CF_RMS from step 50 is less than three(step 52) and no peaks were discarded from the statistical analysis ofthe maximum peaks in Deriv>0_WF (step 70), then the method proceeds toevaluate the magnitude of %_Mean_Fltd_Max_PK determined in step 68. If%_Mean_Fltd_Max_PK is less than or equal to ten (which meansMax_Threshold and Min_Threshold are reasonably close in value) (step72), then:

Thrsh10=Min(Max_Threshold andMin_Threshold)−2*(|Max_Threshold−Min_Threshold|)(step 74).

If %_Mean_Fltd_Max_PK is greater than ten (step 72), then:

Thrsh10=Min(Max Threshold and Min Threshold)−(|Max Threshold−MinThreshold|)(step 76)

and the method proceeds to step 78.

If %_Diff_Threshold (calculated at step 68) is greater than 40 (step78), then

${Threshold} = {{Mid\_ Threshold} = \frac{\begin{matrix}{({Threshold\_ multiplier})*} \\\left( {{Max\_ Threshold} + {Min\_ Threshold}} \right)\end{matrix}}{2}}$

where Threshold_multiplier is a user-selectable value between 0 and 1(step 80). In preferred embodiments, the value used is 1. If%_Diff_Threshold (calculated at step 68) is less than or equal to 40(step 78), then the method proceeds to step 82.

If %_Mean_Fltd_Max_PK is greater than or equal to 0.5 (step 82), then

Threshold=Thrsh10 (step 84).

If %_Mean_Fltd_Max_PK is less than 0.5 (step 82), then

Threshold=0.67*Min(Max_Threshold and Min_Threshold) (step 86).

Generally, %_Mean_Fltd_Max_PK is less than 0.5 when data is very steadyand the values of the peaks are all about the same amplitude.

Referring back to step 52 of FIG. 3, if Adj/Filter_CF_RMS is less thanthree and peaks were discarded from the statistical analysis of themaximum peaks in Deriv>0_WF (step 70), then the method proceeds to step88. If %_Diff_Threshold is less than or equal to 40 (step 88), then

Threshold=0.5×MinΔ+Min_base_value (step 90).

If %_Diff_Threshold is greater than 40 (step 88), then

Threshold=Min_Threshold (step 92).

It should be noted that if peaks are discarded, then there are peaks inthe original “number of peaks for analysis” which are statisticaloutliers (values greater than μ±nσ).

Referring again to step 52 of FIG. 3, if Adj/Filter_CF_RMS is greaterthan or equal to three and no peaks were discarded from statisticalanalysis of the maximum peaks in Deriv>0_WF, then the Deriv>0_WFwaveform is used as the tachometer signal and to set the threshold (step54) according to:

Threshold=μ+2σ (step 94)

where μ and σ are calculated from the N peaks of the Deriv>0_WFwaveform.

Tables 1 and 2 provide a summary of the threshold calculation.

TABLE 1 Data Calculations Calculations to determine ThresholdFiltered_WF_CF_RMS (step 20) Deriv_CF_RMS (step 24) Deriv > 0_CF_RMSEquivalent Equivalent Equivalent Values Values Values Differ (step 34)Value Value Value Differ Differ Adj_Deriv > 0_CF_RMS (step 40)$\begin{matrix}\frac{{Adj\_ Deriv} > {0{\_ CF}{\_ RMS}}}{{Filtered\_ WF}{\_ CF}{\_ RMS}} \\\left( {{step}\mspace{14mu} 52} \right)\end{matrix}\quad$ ≧3 <3 ≧3 <3 Percent Difference ≦40% >40% ≦40% >40%Threshold (step 88) Threshold Value μ + 2σ for See Table 2.Mid_Threshold Bad Data 0.5*MinΔ + Min_Threshold Deriv > 0_WF Base Value

TABLE 2 Percent Mean Filtered Max Peak Threshold Value <0.5 0.67 *Min(Max_Threshold and Min_Threshold) ≧0.5 and ≦10 Min of (Max_Thresholdand Min_Threshold) − 2*(|Max_Threshold − Min_Threshold|) >10Min(Max_Threshold and Min_Threshold) − (|Max_Threshold − Min_Threshold|)

Method to Determine the Number (N) of Peaks for Analysis

Following is a preferred embodiment of a routine for determining thenumber of largest peaks in a given waveform necessary to effectivelyevaluate data for determination of a threshold level. To calculate thenumber peaks (N) for analysis:

If (Filter_WF_Kurtosis > Deriv_CF_RMS) AND   (Deriv>0_CF_RMS >Deriv_CF_RMS) AND   (Deriv>0_CF_RMS > Filter_WF_Kurtosis) THEN   PeakMultiplier = 1.5 Else   Peak Multiplier = 0.75 Endif   N = [Integervalue (rounded up) of ((Peak Multiplier) × (Cycles of RPM))] where: Filter_WF_Kurtosis is the kurtosis of the filtered waveform. Kurtosisis an indication of the    shape of the distribution of data. A value of3 represents a normal distribution.    Values less than 3 indicateflatter distributions. Values greater than 3 indicate a    sharper (morenarrow) distribution. Deriv_CF_RMS is the crest factor calculated fromthe waveform produced by taking the    derivative of the filteredwaveform (Deriv_WF) as calculated above in step 24. Deriv>0_CF_RMS isthe crest factor calculated form the waveform produced from derivative   values greater than zero (Deriv>0_WF) as calculated above in step 34.Peak Multiplier is the value used to scale the final number of peaks foranalysis. Cycles of RPM is the number of RPM cycles present in a givenwaveform, calculated as:    Cycles of RPM = Sampling Rate (sec) × (RatedSpeed (RPM) / 60)

Methods for Sorting Out Statistical Outliers

The following routine takes an array of data values and discards valuesoutside the statistically calculated boundaries. In a preferredembodiment, there are four methods or criteria for setting theboundaries.

Method 1: Non-Conservative, Using Minimum and Maximum StatisticalBoundaries

Consider an array of P values (or elements) where P₀ represents thenumber of values in the present array under evaluation. Now let P⁻¹represent the number of values in the array evaluated a single stepbefore P₀, let P⁻² represent the number of values in the array evaluateda single step before P⁻¹, and let P⁻³ represent the number of values inthe array evaluated a single step before P⁻².

Step 1:

While evaluating the array of values for either the first time or P₀ ≠P⁻¹,  {  Calculate the mean (μ) and standard deviation (σ) for P₀  ${{{If}\mspace{14mu} \frac{n\; \sigma}{\mu}} \geq 0.1},{{then}\mspace{101mu} \left( {{n = 1},{2\mspace{14mu} {or}\mspace{14mu} 3}} \right)}$  Include array values such that   μ − nσ < values < μ + nσ  Else  STOP, values are within statistical boundaries.  Endif  }

Step 2:

  If P₀ = P⁻¹, then  While P⁻¹ ≠ P⁻², and P₀ = P⁻¹   {   Calculate themean (μ) and standard deviation (σ) for P₀    $\begin{matrix}{{{{If}\mspace{14mu} \frac{n\; \sigma}{2\mu}} \geq 0.1},{{then}\mspace{101mu} \left( {{n = 1},{2\mspace{14mu} {or}\mspace{14mu} 3}} \right)}} \\{\mspace{31mu} {{Include}\mspace{14mu} {array}\mspace{14mu} {values}\mspace{14mu} {such}\mspace{14mu} {that}}} \\{\mspace{31mu} {{\mu - \frac{n\; \sigma}{2}} < {values} < {\mu + \frac{n\; \sigma}{2}}}}\end{matrix}\quad$   Else    STOP, values are within statisticalboundaries.   Endif   }  Endif

Step 3:

If P₀ = P⁻¹ = P⁻², and P⁻² ≠ P⁻³, then   Calculate the mean (μ) andstandard deviation (σ) for P₀   Include array values such that   0.9μ <values < 1.1μ Else   STOP, values are within statistical boundaries.Endif

Method 2: Non-Conservative, Using Maximum Statistical Boundary Only (NoMinimum Boundary)

Use the same procedure as in Method 1 except only values exceeding theupper statistical boundaries are discarded. The minimum boundary is setto zero.

Method 3: Conservative, Using Minimum and Maximum Statistical Boundaries

Discard values based on Method 1, Step 1 only.

Method 4: Conservative, Using Maximum Statistical Boundary Only (NoMinimum Boundary)

Discard values based on Method 1, Step 1 only and based on valuesexceeding the upper statistical boundaries. The minimum boundary is setto zero.

Example of Method 1 for Sorting Out Statistical Outliers

As an example of the sorting Method 1, consider an original set ofvalues, P₀, containing the 21 values listed below in Table 3 below, withn=1.

TABLE 3 0.953709 0.828080 0.716699 0.653514 0.612785 0.582031 0.5792090.557367 0.545801 0.495215 0.486426 0.486053 0.475123 0.472348 0.4671290.465488 0.446327 0.440497 0.437959 0.427256 0.411627

The mean (μ) of this original set, P₀, is 0.54955 and standard deviation(σ) is 0.13982. Therefore, in Step 1 of Method 1,

$\frac{n\; \sigma}{\mu} = {{1*\frac{0.13982}{0.54955}} = {0.25442.}}$

Since 0.25442 is greater than 0.1, calculate

μ−nσ=0.54955−1*0.13982=0.409735

and

ν+nσ=0.54955+1*0.13982=0.689373.

Next, define the set P⁻¹=P₀ and define a new set P₀, the values of whichare all the values of P⁻¹ that are between the values μ+σ=0.689343 andμ−σ=0.409735. The set P₀ now contains the values listed below in Table4, wherein three outlier values have been eliminated.

TABLE 4 0.653514 0.612785 0.582031 0.579209 0.557367 0.545801 0.4952150.486426 0.486053 0.475123 0.472348 0.467129 0.465488 0.446327 0.4404970.437959 0.427256 0.411627

Since P₀≠P⁻¹, Step 1 is repeated, where for the set P₀:

μ=0.50234,

σ=0.06946,

σ/μ=0.138263,

μ+σ=0.571797, and

μ−σ=0.432887.

Now define the set P⁻²=P⁻¹, and P⁻¹=P₀ and define a new set P₀, thevalues of which are all the values of P⁻¹ that are between the valuesμ+σ=0.571797 and μ−σ=0.432887. The set P₀ now contains the values listedbelow in Table 5, wherein four more outlier values have been eliminated.

TABLE 5 0.557367 0.545801 0.495215 0.486426 0.486053 0.475123 0.4723480.467129 0.465488 0.446327 0.440497 0.437959 0.427256 0.411627

Since P₀≠P⁻¹, Step 1 is repeated, where for the set P₀:

μ=0.472473,

σ=0.041332, and

σ/μ=0.087481.

Since

σ/μ=0.087481≦1,

all the members of the array P₀ are statistically close in value andneed no more sorting.

If at any point in the calculations P₀=P⁻¹ and P⁻¹≠P⁻², then Step 2would be executed instead of Step 1. In the example above, since P₀≠P⁻¹for every iteration, only Step 1 was necessary for the calculations.

The foregoing description of preferred embodiments for this inventionhave been presented for purposes of illustration and description. Theyare not intended to be exhaustive or to limit the invention to theprecise form disclosed. Obvious modifications or variations are possiblein light of the above teachings. The embodiments are chosen anddescribed in an effort to provide the best illustrations of theprinciples of the invention and its practical application, and tothereby enable one of ordinary skill in the art to utilize the inventionin various embodiments and with various modifications as are suited tothe particular use contemplated. All such modifications and variationsare within the scope of the invention as determined by the appendedclaims when interpreted in accordance with the breadth to which they arefairly, legally, and equitably entitled.

1. A method for deriving information from a tachometer signal, themethod comprising: (a) sensing the tachometer signal; (b) determining acharacteristic of the tachometer signal; (c) taking a running derivativeof the tachometer signal to generate a derivative signal; (d)determining a characteristic of the derivative signal; (e) calculating acharacteristic ratio, which is a ratio of the characteristic of thederivative signal to the characteristic of the tachometer signal; and(f) comparing the characteristic ratio to a predetermined value.
 2. Themethod of claim 1 further comprising performing one of the followingbased on the comparing of step (f): (g1) using the tachometer signal toderive a threshold value for tachometer pulses in the tachometer signaland extracting tachometer pulses from the tachometer signal based on thethreshold value; (g2) using the derivative signal to derive a thresholdvalue for tachometer pulses and extracting tachometer pulses from thederivative signal based on the threshold value; and (g3) determiningthat a meaningful threshold value cannot be derived due to insufficientinformation in the tachometer signal.
 3. The method of claim 2 wherein:step (g1) is performed if the characteristic ratio is less than thepredetermined value; and step (g2) is performed if the characteristicratio is not less than the predetermined value.
 4. The method of claim 1wherein the characteristic of the tachometer signal is a crest factor ofthe tachometer signal, the characteristic of the derivative signal is acrest factor of the derivative signal, and the characteristic ratio is aratio of the crest factor of the derivative signal to the crest factorof the tachometer signal.
 5. The method of claim 4 wherein thepredetermined value is
 3. 6. The method of claim 1 further comprisingthe following steps after step (c) and before step (d): (g) determininga mean value and a standard deviation value of the derivative signal;(h) setting all negative values in the derivative signal to zero togenerate an all-positive-value derivative signal; (i) finding N numberof peaks in the all-positive-value derivative signal; and (j) discardingstatistical outlier peaks from the N number of peaks in theall-positive-value derivative signal, wherein the statistical outlierpeaks have an amplitude greater than a sum of the mean value and thestandard deviation value, wherein step (d) comprises determining acharacteristic of the all-positive-value derivative signal from whichthe outlier peaks have been discarded.
 7. The method of claim 6 furthercomprising: (k) based on the comparing of step (f), determining to usethe derivative signal to derive a threshold value for tachometer pulsesin the derivative signal; and (l) if statistical outlier peaks werediscarded in step (j), determining that a meaningful threshold valuecannot be derived due to insufficient information in the tachometersignal.
 8. A method for deriving information from a tachometer signal,the method comprising: (a) sensing the tachometer signal; (b) taking arunning derivative of the tachometer signal to generate a derivativesignal; (c) determining a mean value and a standard deviation value ofthe derivative signal; (d) setting all negative values in the derivativesignal to zero to generate an all-positive-value derivative signal; (e)finding N number of peaks in the all-positive-value derivative signal;(f) discarding statistical outlier peaks from the N number of peaks inthe all-positive-value derivative signal, wherein the statisticaloutlier peaks have an amplitude greater than a sum of the mean value andthe standard deviation value; (g) determining a characteristic of theall-positive-value derivative signal from which the statistical outlierpeaks have been discarded; and (h) using the characteristic to determineone of the following: the tachometer signal is to be used to derive athreshold value for tachometer pulses in the tachometer signal; thederivative signal is to be used to derive a threshold value fortachometer pulses in the derivative signal; and a meaningful thresholdvalue cannot be derived due to insufficient information in thetachometer signal.
 9. A method for deriving information from atachometer signal, the method comprising: (a) sensing the tachometersignal; (b) taking a running derivative of the tachometer signal togenerate a derivative signal; (c) determining statisticalcharacteristics of the tachometer signal and the derivative signal; and(d) deriving a threshold value for tachometer pulses in the tachometersignal based on the statistical characteristics of the tachometer signaland the derivative signal.